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Where can I find a calculus textbook that emphasizes differentials? Is there such a book that I could realistically require my calculus students to use?

I want a textbook that supports me when I tell my students something like:

$\Delta((x^2+1)^5)\approx5(x^2+1)^4\Delta(x^2+1)\approx5(x^2+1)^4(2x\Delta x)$

$d((x^2+1)^5)=5(x^2+1)^4d(x^2+1)=5(x^2+1)^4(2x\ dx)$


$\Sigma_{k=1}^n 3x_k^2\Delta x_k\approx\Sigma_{k=1}^n\Delta(x_k^3)=x_n^3-x_1^3$

$\int_{x=0}^{x=4}3x^2\ dx=\int_{x=0}^{x=4}d(x^3)=4^3-0^3=64$

Perhaps I could write this book someday, but it'd be a lot easier for me if my students and I could just buy and/or download a book that takes this approach without neglecting to provide a cornucopia of exercises, examples, and applications similar to what's available in today's most popular calculus textbooks.

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(btw, I think you can just present to your students the manipulatons with differentials as mere notation, emphasizing that in the future -when they'll learn differential forms- they will see it was not just an abuse of language after all) –  Qfwfq Mar 11 '11 at 19:48
I actually tried Keisler's book last year, and of the books I've seen, it's approach to differentials is by far the closest to what I want (though it doesn't go quite as far as I want to go). However, compared to today's textbooks, the applications feel dated, and the illustrations are too few and too primitive. (It was published in 1976.) Also, my students said it was a very difficult read; I think a partial cause was Keisler inserting more rigor than appropriate for my students, who are mostly engineering majors. –  David Milovich Mar 11 '11 at 22:30
Maybe we could crowd-source a rewrite of Keisler's book. –  Kevin O'Bryant Apr 4 '11 at 2:19
Our dept uses Stewart as a mandatory Calc text, but I teach the differentiation chapters from a differentials point of view. (one of Devian Tray's former students turned me on to the idea.) The only problem is when students try to do the homework, they get confused by the traditional approach that Stewart uses. To that end, I'm planning on writing companion materials for Chapters 3 & 4 of the Stewart text. It won't be a full text in any sense, but if you're interested, send me your email address and I'll send you a pdf copy when it's completed. –  Aeryk Apr 4 '11 at 4:45

8 Answers 8

up vote 25 down vote accepted

There is a marvelous old book (19th Century if I recall correctly) where I learned Calculus the first time, called "Calculus Made Easy" by Silvanus P. Thompson, and subtitled "What one fool can do another can". He explains that dx means a "little bit of x" and shows a square with sides x and x + dx and you can see why you can "ignore dx^2". Of course it isn't rigorous in any sense, but it uses differentials to get all the essential ideas of both differential and integral Calculus across quickly and smoothly. Needless to say, once I had absorbed all these essential ideas I went on to read more rigorous books where limits were introduced and used to make precise what I already understood well from this intuitive introduction. If I recall correctly Calculus Made Easy was republished some years back (Dover?) and was quite popular. I would suggest that you recommend it to your students, with appropriate caveats.

(Added later) I checked online and indeed there is a recent reprinting (available from Amazon and the other usual places). Moreover it has three new chapters written by the late great Martin Gardner aimed at the modern reader. I'm going to buy myself a copy!

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And of course it can be made perfectly rigorous through sheaf theory, provided that the site includes "objects of nilpotent infinesimals". This point of view is developed in synthetic differential geometry, of which there are a number of accounts. The book by Moerdijk and Reyes is one, and I couldn't help thinking of this because they have an interesting discussion of a theorem due to Ambrose, Palais, and Singer (on an equivalence between symmetric connections and sprays). :-) –  Todd Trimble Mar 11 '11 at 20:27
I have always liked this book and wish modern calculus textbooks were more like this. –  Deane Yang Mar 11 '11 at 21:45
Thank you. Searching for "Calculus Made Easy" on Amazon also pointed me towards a revised edition co-authored by Martin Gardner. –  David Milovich Mar 11 '11 at 22:40
The 1914 edition is free online from Project Gutenberg: –  Toby Bartels Apr 3 '11 at 23:53
Update: I read Gardner-Thompson. What it explains it explains excellently, but the book feels dated in the same ways Keisler's book does, and when I teach a class in my university's 3-semester calculus sequence, I'm expected to cover many topics not in this book. (Also, some of these missing topics I really do want to cover.) However, I will certainly recommend this book as a supplemental text the next time I teach calculus. The conceptual exposition is extremely reader-friendly. (However, any student with a weak pre-calculus foundation will frequently get lost in the examples.) –  David Milovich Apr 4 '11 at 16:46

Henle and Kleinberg's Infinitesimal Calculus is available in paperback for less than ten dollars and the reviews at Amazon are very strong, at least for use as a supplemental text.

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I have written a textbook called Intuitive Infinitesimal Calculus, which teaches infinitesimal calculus the classical, informal way, informed by my Ph.D. research on the history of the Leibnizian calculus.

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This approach is suggested by Tevian Dray and Corinne Manogue in their program of Bridging the Vector Calculus Gap. They focus on multivariable calculus and differential forms, but they discuss single-variable calculus (pdf) once. Unfortunately, they don't seem to have a textbook for that.

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Thanks for the link. Dray, Manogue, and I want the same thing. –  David Milovich Apr 4 '11 at 13:50

Does Vakil's book "real analysis through modern infinitesimals" contain the additional material you are looking for?

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I could see myself someday using a book like this in the context of teaching graduate students or very strong undergrads. However, I was looking for a friendlier book for the purpose of teaching calculus with infinitesimals. –  David Milovich Apr 19 '13 at 21:50

If what you are looking for is "multivariable calculus with infinitesimals", you might be interested in Stroyan's book at The table of contents is here:

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Calculus by M.E. Munroe stresses the differential and the Leibniz notation.

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